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Extremes

, Volume 18, Issue 1, pp 99–108 | Cite as

On probability of high extremes for product of two independent Gaussian stationary processes

  • Vladimir I. Piterbarg
  • Alexander Zhdanov
Article

Abstract

Let X(t), Y(t), t≥0, be two independent zero-mean stationary Gaussian processes, whose covariance functions are such that r i (t)=1−|t| a i +o(|t| a i ) as \(t\rightarrow 0\), with 0<a i ≤2, i=1,2 and both of the functions are less than one for non-zero t. We derive for any p the exact asymptotic behavior of the probability \(P(\max _{t\in \lbrack 0,p]}X(t)Y(t)>u)\) as \(u\rightarrow \infty \). We discuss possibilities generalizing obtained results to other Gaussian chaos processes h(X(t)), with a Gaussian vector process X(t) and a homogeneous function h of positive order.

Keywords

Gaussian processes Gaussian chaos High extremes probabilities Double sum method 

AMS 2000 Subject Classifications

Primary 60G15 Secondary 60K30 60K40 60G70 

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References

  1. Hashorva, E., Korshunov, D.A., Piterbarg, V.I.: On extremal behavior of gaussian chaos. Dokl. Math. 88, 566–568 (2013)CrossRefMATHMathSciNetGoogle Scholar
  2. Hashorva, E., Korshunov, D.A., Piterbarg, V.I.: Extremal Behavior of Gaussian Chaos via Geometric Approach. Extremes, submitted (2014)Google Scholar
  3. Piterbarg, V.I.: Asymptotic Methods in the Theory of Gaussian Processes and Fields. In: Transl. Math. Monographs, vol. 148. AMS, Providence, RI (1996)Google Scholar
  4. Piterbarg, V.I.: High excursions for nonstationary generalized chi-square processes. Stoch. Process Appl. 53, 307–337 (1994)CrossRefMATHMathSciNetGoogle Scholar
  5. Piterbarg, V., Stamatovic, S.: Limit Theorem for High Level a-Upcrossings by χ-Process. Theor. Probab. Appl. 48(4), 734–741 (2004)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Moscow Lomonosov State UniversityMoscowRussia

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